Question
All the numbers from $19$ to $93$ are written consecutively to form the number $N =19202122...........919293$. Find the largest power of $3$ that divides $N$.
The following hint has been provided.
The square of any integer is either divisible by $4$ or leaves a remainder $1$ divided by $4$. Thus, an integer which leaves a remainder $2$ or $3$ when divided by $4$ can never be a square. If a prime p divides a square number the $p^2$ also divides that number.
On
With a bit more rigour and without a calculator, we have:
$$N =\sum_{k=0}^{93-19 } (93-k)\cdot 10^{2k}$$
Now if we inspect this $\bmod 9$ we can use the fact that $10^j \equiv 1 \mod 9$ to find:
$$N \equiv\sum_{k=0}^{74} (93-k) \mod 9$$ $$N \equiv 75\cdot 93 - \sum_{k=0}^{74} k \mod 9$$ $$N \equiv 3\cdot 3 - \frac{1}{2}\cdot 74\cdot 75 \mod 9$$ $$N \equiv 9 - 37\cdot 3 \mod 9$$ $$N \equiv 9 - 1\cdot 3 \mod 9$$ $$N \equiv 6 \mod 9$$
Thus we know that $N$ is not divisible by $9$. But since $(N \bmod 9) \equiv N \mod 3$ we can re-use the above calculation to immediately find that $N \equiv 0 \mod 3$ and thus is divisible by $3$. That is the largest power of $3$ that divides $N$.
First of all calculate sum of digits that sum of natural numbers from 19 to 93 which is 4200. It is divisible by 3 but not 9. For divisibility of higher power it has to be divisible by 9 but it is not divisible by 9. So higher power should be 1.